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Inverse Littlewood-Offord Theorems and the Condition Number Of Annals of Mathematics, 169 (2009), 595{632 Inverse Littlewood-Offord theorems and the condition number of random discrete matrices By Terence Tao and Van H. Vu* Abstract Consider a random sum η1v1 + ··· + ηnvn, where η1; : : : ; ηn are indepen- dently and identically distributed (i.i.d.) random signs and v1; : : : ; vn are inte- gers. The Littlewood-Offord problem asks to maximize concentration probabil- ities such as P(η1v1+···+ηnvn = 0) subject to various hypotheses on v1; : : : ; vn. In this paper we develop an inverse Littlewood-Offord theory (somewhat in the spirit of Freiman's inverse theory in additive combinatorics), which starts with the hypothesis that a concentration probability is large, and concludes that almost all of the v1; : : : ; vn are efficiently contained in a generalized arithmetic progression. As an application we give a new bound on the magnitude of the least singular value of a random Bernoulli matrix, which in turn provides upper tail estimates on the condition number. 1. Introduction Let v be a multiset (allowing repetitions) of n integers v1; : : : ; vn. Consider a class of discrete random walks Yµ,v on the integers Z, which start at the origin and consist of n steps, where at the ith step one moves backwards or forwards with magnitude vi and probability µ/2, and stays at rest with probability 1−µ. More precisely: Definition 1.1 (Random walks). For any 0 ≤ µ ≤ 1, let ηµ 2 {−1; 0; 1g denote a random variable which equals 0 with probability 1 − µ and ±1 with probability µ/2 each. In particular, η1 is a random sign ±1, while η0 is iden- tically zero. Given v, we define Yµ,v to be the random variable n X µ Yµ,v := ηi vi i=1 *T. Tao is a Clay Prize Fellow and is supported by a grant from the Packard Foundation. V. Vu is an A. Sloan Fellow and is supported by an NSF Career Grant. 596 TERENCE TAO AND VAN H. VU µ µ where the ηi are i.i.d. copies of η . Note that the exact enumeration v1; : : : ; vn of the multiset is irrelevant. The concentration probability Pµ(v) of this random walk is defined to be the quantity (1) Pµ(v) := max P(Yµ,v = a): a2Z Thus we have 0 < Pµ(v) ≤ 1 for any µ, v. The concentration probability (and more generally, the concentration function) is a central notion in probability theory and has been studied exten- sively, especially by the Russian school (see [21], [19], [18] and the references therein). The first goal of this paper is to establish a relation between the magnitude of Pµ(v) and the arithmetic structure of the multiset v = fv1; : : : ; vng. This gives an answer to the general question of finding conditions under which one can squeeze large probability inside a small interval. We will primarily be interested in the case µ = 1, but for technical reasons it will be convenient to consider more general values of µ. Generally, however, we think of µ as fixed, while letting n become very large. A classical result of Littlewood-Offord [16], found in their study of the number of real roots of random polynomials, asserts that if all of the vi's −1=2 are nonzero, then P1(v) = O(n log n). The log term was later removed by Erd}os[5]. Erd}os'bound is sharp, as shown by the case v1 = ··· = vn 6= 0. How- ever, if one forbids this special case and assumes that the vi's are all distinct, then the bound can be improved significantly. Erd}osand Moser [6] showed −3=2 that under this stronger assumption, P1(v) = O(n ln n). They conjectured that the logarithmic term is not necessary and this was confirmed by S´ark¨ozy and Szemer´edi[22]. Again, the bound is sharp (up to a constant factor), as can be seen by taking v1; : : : ; vn to be a proper arithmetic progression such as 1; : : : ; n. Later, Stanley [24], using algebraic methods, gave a very explicit bound for the probability in question. The higher dimensional version of Littlewood-Offord’s problem (where d the vi are nonzero vectors in R , for some fixed d) also drew lots of attention. Without the assumption that the vi's are different, the best result was obtained by Frankl and F¨urediin [7], following earlier results by Katona [11], Kleitman [12], Griggs, Lagarias, Odlyzko and Shearer [8] and many others. However, the techniques used in these papers did not seem to yield the generalization of S´ark¨ozyand Szemer´edi'sresult (the O(n−3=2) bound under the assumption that the vectors are different). The generalization of S´ark¨ozyand Szemer´edi'sresult was obtained by Hal´asz[9], using analytical methods (especially harmonic analysis). Hal´asz' paper was one of our starting points in this study. INVERSE LITTLEWOOD-OFFORD AND CONDITION NUMBER 597 In the above two examples, we see that in order to make Pµ(v) large, we have to impose a very strong additive structure on v (in one case we set the vi's to be the same, while in the other we set them to be elements of an arithmetic progression). We are going to show that this is the only way to make Pµ(v) large. More precisely, we propose the following phenomenon: If Pµ(v) is large, then v has a strong additive structure. In the next section, we are going to present several theorems supporting this phenomenon. Let us mention here that there is an analogous phenomenon in combinatorial number theory. In particular, a famous theorem of Freiman asserts that if A is a finite set of integers and A+A is small, then A is contained efficiently in a generalized arithmetic progression [28, Ch. 5]. However, the proofs of Freiman's theorem and those in this paper are quite different. As an application, we are going to use these inverse theorems to study µ random matrices. Let Mn be an n by n random matrix, whose entries are i.i.d. copies of ηµ. We are going to show that with very high probability, µ the condition number of Mn is bounded from above by a polynomial in n (see Theorem 3.3 below). This result has high potential of applications in the theory of probability in Banach spaces, as well as in numerical analysis and theoretical computer science. A related result was recently established by Rudelson [20], with better upper bounds on the condition number but worse probabilities. We will discuss this application with more detail in Section 3. To see the connection between this problem and inverse Littlewood-Offord theory, observe that for any v = (v1; : : : ; vn) (which we interpret as a column µ vector), the entries of the product Mn v are independent copies of Yµ,v. Thus T µ we expect that v is unlikely to lie in the kernel of Mn unless the concentration probability Pµ(v) is large. These ideas are already enough to control the µ singularity probability of Mn (see e.g. [10], [25], [26]). To obtain the more quantitative condition number estimates, we introduce a new discretization technique that allows one to estimate the probability that a certain random variable is small by the probability that a certain discretized analogue of that variable is zero. The rest of the paper is organized as follows. In Section 2 we state our main inverse theorems. In Section 3 we state our main results on condition numbers, as well as the key lemmas used to prove these results. In Section 4, we give some brief applications of the inverse theorems. In Section 7 we prove the result on condition numbers, assuming the inverse theorems and two other key ingredients: a discretization of generalized progressions and an extension of the famous result of Kahn, Koml´osand Szemer´edi[10] on the probability that a random Bernoulli matrix is singular. The inverse theorems is proven in Section 6, after some preliminaries in Section 5 in which we establish basic properties of Pµ(v). The result about discretization of progressions are proven 598 TERENCE TAO AND VAN H. VU in Section 8. Finally, in Section 9 we prove the extension of Kahn, Koml´osand Szemer´edi[10]. We conclude this section by setting out some basic notation. A set 0 P = fc + m1a1 + ··· + mdadjMi ≤ mi ≤ Mi g is called a generalized arithmetic progression (GAP) of rank d. It is convenient to think of P as the image of an integer box 0 B := f(m1; : : : ; md)jMi ≤ mi ≤ Mi g in Zd under the linear map Φ:(m1; : : : ; md) 7! c + m1a1 + ··· + mdad: The numbers ai are the generators of P . In this paper, all GAPs have ra- tional generators. A GAP is proper if Φ is one to one on B. The product Qd 0 0 i=1(Mi − Mi + 1) is the volume of P . If Mi = −Mi and c = 0 (so P = −P ) then we say that P is symmetric. For a set A of reals and a positive integer k, we define the iterated sumset kA := fa1 + ··· + akjai 2 Ag: One should take care to distinguish the sumset kA from the dilate k·A, defined for any real k as k · A := fkaja 2 Ag: We always assume that n is sufficiently large. The asymptotic notation O(), o(), Ω(), Θ() is used under the assumption that n ! 1. Notation such as Od(f) means that the hidden constant in O depends only on d. 2. Inverse Littlewood-Offord theorems Let us start by presenting an example when Pµ(v) is large.
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